2.9 KiB
Derivatives
Let \mathrm{M} be a differential manifold with \dim \mathrm{M} = n \in \mathbb{N} used throughout the section. Let \mathrm{TM} and \mathrm{T^*M} denote the tangent and cotangent bundle, V and V^* the fiber and dual fiber bundle and \mathscr{B} the tensor fiber bundle.
Lie derivative
Definition 1: the Lie derivative on a section of a tangent bundle
\mathscr{L}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to \Gamma(\mathrm{TM})is a map defined by
\mathscr{L}_\mathbf{w} \mathbf{v} = \mathbf{w} \circ \mathbf{v} - \mathbf{v} \circ \mathbf{w} = [\mathbf{w}, \mathbf{v}],for all
\mathbf{w}, \mathbf{v} \in \Gamma(\mathrm{TM}).
In which the bracket formulation is also referred to as the Lie bracket.
Proposition 1: the Lie derivative can be decomposed into
\mathscr{L}\mathbf{w} \mathbf{v} = \mathscr{L}\mathbf{w}^i \mathbf{v} \partial_i = (w^j \partial_j v^i - v^j \partial_j w^i) \partial_i,for all
\mathbf{w}, \mathbf{v} \in \Gamma(\mathrm{TM}).
??? note "Proof:"
Will be added later.
Exterior derivative
Definition 2: the exterior derivative
d: \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big) \to \Gamma \big(\bigwedge_{k+1}(\mathrm{T}\mathrm{M}) \big)of a $k$-form field,k \in \mathbb{N}[k \leq n]is the $(k+1)$-form field
\begin{align*} d \bm{\omega} &= d \omega_{|i_1 \dots i_k|} \wedge dx^{i_1} \wedge \dots \wedge dx^{i_k}, \ &= \partial_j \omega_{|i_1 \dots i_k|} dx^j \wedge dx^{i_1} \wedge \dots \wedge dx^{i_k}, \end{align*}for all
\bm{\omega} \in \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big).
From the definition of the exterior definition the following results arises.
Theorem 1: we have that
\forall\bm{\omega} \in \Gamma \big(\bigwedge_n(\mathrm{T}\mathrm{M}) \big): d \bm{\omega} = \mathbf{0},\forall\bm{\omega} \in \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big), k \in \mathbb{N}[k \leq n]: d^2 \bm{\omega} = \mathbf{0}.
??? note "Proof:"
Will be added later.
Hodge star operator
Definition 3: the hodge star operator
*: \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big) \to \Gamma \big(\bigwedge_{n-k}(\mathrm{T}\mathrm{M}) \big)withk \in \mathbb{N}[k \leq n]has the following properties
\forall \bm{\omega} \in \Gamma \big(\bigwedge_0(\mathrm{T}\mathrm{M}) \big): * \bm{\omega} = \bm{\epsilon},* (dx^{i_1} \wedge \dots \wedge dx^{i_k}) = \bm{\epsilon} \lrcorner \mathbf{g}^{-1}(dx^{i_1}) \lrcorner \dots \lrcorner \mathbf{g}^{-1}(dx^{i_k}),for all
dx^{i_1} \wedge \dots \wedge dx^{i_k} \in \Gamma \big(\bigwedge_k(\mathrm{T}\mathrm{M}) \big)with\bm{\epsilon}the Levi-Civita tensor\bm{\epsilon} \in \big(\bigwedge_n(\mathrm{T}\mathrm{M}) \big)and\mathbf{g}^{-1}: \Gamma(\mathrm{T}^*\mathrm{M}) \to \Gamma(\mathrm{T}\mathrm{M})the dual metric.